If you attached the signal line to and Arduino input pin (say through a 1K resistor) and then used a 10K pull-down to ground then I cannot see what issues you can have.

The max current that would flow to ground through the 10K would be 5/10,000 = 0.5mA (actually it would be 5/11,000 if you used a 1K). This is not likely to cause a problem to anything. If you chose to connect directly (rather than through a resistor) and then for some mad reason you made the pin an output and set it low then you could cause problems. Otherwise, I can't see any issues.

On the other hand, why make it so complicated?If the 7805 is just there to make the 9V into 5V for the arduino to read,I would ditch it and the capacitors and just use two 5K resistors to make the 9V into 4.5V, and when 9V goes away the 4.5V goes away too.

Plus the other regulator is connected to a battery charger, and I had problems in the past using the same regulator for the signal line.Seems like even after power goes out, for whatever reason the battery charger will leak backward voltage.

Plus the L78L05 is a TO-92 which is a small form factor which supplies the SIGNAL line.Plus I have over 20 of the L78L05's laying around, lol

You could consider a higher value resistor. Maybe 50K. This reduces the waste current (and generates less heat). Useful if power is at a premium (e.g., small transformer), but probably still not a big deal even at 10K.

Highest value? There will be limiting factors to that. Basically, it has to be low enough that it can't be overcome spuriously by noise or external factors. I imagine 100K would be fine in most cases, but it would be best to check first.

Lowest value is dictated by acceptable waste current (lower values drain more power from Vcc to Gnd, and will eventually act as a short for extremely low resistance) and the ability of your trigger to overcome the pull-down.

I bought an order of 100 51K-ohm resistors (SMT, order-stuffer in one of my Mouser orders) for my projects to use as pullups/pulldowns ... Works fine. I think the general idea is that going well above ~100Kohm (especially >1 megaohm) makes the transient effects of RF and noise in the circuit/processor significant enough to possibly cause false positive readings on the line, so something in the <=100K ohm range seems reasonable.